the value that has to be the
same in all logarithmic expressions
and b can not 1
3
common logarithm
it is a log with a base of 10; sometimes
written as just "log"
4
exponential equation
an equation that contains more than one
exponential expression
5
inverse function
the function that is the result of switching
the input values with the output.
6
logarithmic equation
an equation that has a logarithm of a
value
7
logarithmic function
a function of y = logb x, where b > 1;
and the inverse is y = b
8
natural logarithm
a logarithm that has a base of e, written
as ln
8.1 Exponential Growth (pg. 465 - 473)To graph an exponetial fuction you will need to make a table of values. There will be y=a2^x graphs and y=ab^x. In both of these a>0 and b>1. In a graph of y=ab^(x-h)+k, h will be the horizontal shift and k will be the vertical shift. Also you may need to find growth factor by using the function y=a(1+r)^t, where a is initial amount and r is the percent increase and t is time.
Real Life application- Exponential Growth can be used in real life situations like population growth over time. Or a banking account that shows amount of interest over a specific number of year
Practice Problems:
Identify y-intercept & asymptote of the graph of the function.
1. y=5^x
2. y= 3 · 2^x-1
Graph the function, state the domain and range
1. y= 4 · 5^x-1
3. y= 4 · 2^x-3 +1
8.2 Exponential Decay (pg. 474 - 479)
To graph exponential decay you will also use a table of values. you will use the same type of functions but most likely the b will be a fraction. the shifts are the same but the
decay factor is not; the fuction for that is y=a(1-r)^t but a, r and t are the same thing.
Real Life Application- Exponential Decay can be used in science. It can be used to find the half life of compounds and radioactive elements
Practice Problems:
Tell whether the function is growth or decay.
1. f(x)= 8 · 7^-x
2. f(x)= 5(1/8)^-x
Graph and state domain and range
1. y= (1/3)^x-2
2. y= (1/3)^x -2
8.3 The Number e (pg. 480 - 485)
"e" is a irrational, non-terminating, non-repeating decimal (like pi) that rounded equals 2.72. It is used in natural logarithms, which is like a common log but instead of base 10 the base is "e". to simplify use the properties of logs.
Continuously Compounded Interest Formula: A=Pe^rt
Real Life Application- e can be used in finding compound interest. The more often the person
compounds their money (monthly, weekly, daily, hourly,) the closer their interest rate gets to approaching e.
Practice Problems:
Simplify.
1. e^-2 · 3e^7
2. (4e^-2)^3
Evaulate with calculator.
1. e^1.7
2. -4e^-3
8.4 Logarithmic Functions (pg. 486 - 492)
First see definition at top of page. these functions are the same as an exponential equation.
This is a log: logbaseb a=x
this is a exponential b^x= a
the variables are just in different places.
to solve you either have to change log to exponetial or use this formula loga/logb
when adding log functions you want to multiply and when subtracting you want to divide.
Practice problems:
Evaluate without using a calculator.
1. log7 (343)
2. log4(4^-0.38)
Using inverses, simplify the expression.
1. 35^log35(x)
Find the inverse.
1. y= ln 6x
8.5 Properties of Logarithms (pg. 493 - 500)
Logs have properties similar to exponents. You can see these properties below. Also these properties will help you condense and expand logs.
PROPERTIES
FORMULAS
EXAMPLES
Product Property
logb(x · y) = logb(x) + logb(y).
log10(5 · 20) = log10(5) + log10(20)
x = 2
Quotient Property
loga (x/y) = log
x - loga y
log (15/5) = log 15 - log 5\
x = 0.47712
Power Property
x=blogb(x).=
x=10log4(64)x =
256
Change of Bases
Formula
Example
Practice Problems:
Use change of base formula to evaluate.
1. log 7 (12)
2. log 9 (5/16)
8.6 Solving Exponential and Logarithmic Equations (pg. 501 - 508)
When trying to solve exponential equations, there is a simpler way.
Try getting the bases to the same number. (bx = bx)
Ex: Solve 101–x = 104 1 – x = 4 1 – 4 = x –3 = x
When trying to solve logarithmic equations, there is a simpler way.
Try getting the bases to the same number. (log bx= logbx)
(2,9) (4,20)
9 = ab2
9/b2 =a
20.25 = 9/b2 (b4)
20.25= 9b2
a = 4 b = 1.5
y = 4 (a.5)x
(5,2) (10,6)
2 = a5b
2/5b = a
6 = 2/5b (10b)
3 = 2b
log2 3 = b
a = .156 b = 1.585
y = .156x1.585
Practice Problems:
Write an exponential function of the form y=ab^x that passes through the given points.
1. (1,4), (2,12)
Write a power function of the form y=ax^b that passes through the given points.
1. (5,12), (7,25)
8.8 Logistic Growth Functions (pg. 517 - 522)
Formula
Example
P(3) = 5.808089324
Practice Problems:
Graph and identify asymptotes, y-intercept and point of maximum.
1. y = 5/1+e^-10x
2. y = 8/1+e^-1.02x
Solve.
1. 10/1+2e^-4x= 9
2. 9/1+5e^-0.2x= 3/4
HOW ARE LOGARITHMS RELATED TO REAL LIFE??? =\
VOCABULARY
the viewed value
same in all logarithmic expressions
and b can not 1
written as just "log"
exponential expression
the input values with the output.
value
and the inverse is y = b
as ln
8.1 Exponential Growth (pg. 465 - 473) To graph an exponetial fuction you will need to make a table of values. There will be y=a2^x graphs and y=ab^x. In both of these a>0 and b>1. In a graph of y=ab^(x-h)+k, h will be the horizontal shift and k will be the vertical shift. Also you may need to find growth factor by using the function y=a(1+r)^t, where a is initial amount and r is the percent increase and t is time.
Real Life application- Exponential Growth can be used in real life situations like population growth over time. Or a banking account that shows amount of interest over a specific number of yearPractice Problems:
Identify y-intercept & asymptote of the graph of the function.1. y=5^x
2. y= 3 · 2^x-1
Graph the function, state the domain and range
1. y= 4 · 5^x-1
3. y= 4 · 2^x-3 +1
8.2 Exponential Decay (pg. 474 - 479)
To graph exponential decay you will also use a table of values. you will use the same type of functions but most likely the b will be a fraction. the shifts are the same but the
decay factor is not; the fuction for that is y=a(1-r)^t but a, r and t are the same thing.
Real Life Application- Exponential Decay can be used in science. It can be used to find the half life of compounds and radioactive elements
Practice Problems:
Tell whether the function is growth or decay.
1. f(x)= 8 · 7^-x
2. f(x)= 5(1/8)^-x
Graph and state domain and range
1. y= (1/3)^x-2
2. y= (1/3)^x -2
8.3 The Number e (pg. 480 - 485)
"e" is a irrational, non-terminating, non-repeating decimal (like pi) that rounded equals 2.72. It is used in natural logarithms, which is like a common log but instead of base 10 the base is "e". to simplify use the properties of logs.
Continuously Compounded Interest Formula: A=Pe^rt
Real Life Application- e can be used in finding compound interest. The more often the person
compounds their money (monthly, weekly, daily, hourly,) the closer their interest rate gets to approaching e.
Practice Problems:
Simplify.
1. e^-2 · 3e^7
2. (4e^-2)^3
Evaulate with calculator.
1. e^1.7
2. -4e^-3
8.4 Logarithmic Functions (pg. 486 - 492)
First see definition at top of page. these functions are the same as an exponential equation.
This is a log: logbaseb a=x
this is a exponential b^x= a
the variables are just in different places.
to solve you either have to change log to exponetial or use this formula loga/logb
when adding log functions you want to multiply and when subtracting you want to divide.
Practice problems:
Evaluate without using a calculator.
1. log7 (343)
2. log4(4^-0.38)
Using inverses, simplify the expression.
1. 35^log35(x)
Find the inverse.
1. y= ln 6x
8.5 Properties of Logarithms (pg. 493 - 500)
Logs have properties similar to exponents. You can see these properties below. Also these properties will help you condense and expand logs.
x = 2
x - loga y
x = 0.47712
256
Practice Problems:
Use change of base formula to evaluate.
1. log 7 (12)
2. log 9 (5/16)
8.6 Solving Exponential and Logarithmic Equations (pg. 501 - 508)
When trying to solve exponential equations, there is a simpler way.
Try getting the bases to the same number. (bx = bx)
Ex: Solve 101–x = 104
1 – x = 4
1 – 4 = x
–3 = x
When trying to solve logarithmic equations, there is a simpler way.
Try getting the bases to the same number. (log bx= logbx)
Ex: Solve logb(x2) = logb(2x – 1).
x2 = 2x – 1
x2 – 2x + 1 = 0
(x – 1)(x – 1) = 0
x = 1
Practice Problems:
Solve.
1. 10^x-3 = 100^4x-5
2. 10^2x + 3 = 8
3. 1/4(4)^2x + 1 = 5
8.7 Modeling (pg. 509 - 516)
9 = ab2
9/b2 =a
20.25 = 9/b2 (b4)
20.25= 9b2
a = 4 b = 1.5
y = 4 (a.5)x
2 = a5b
2/5b = a
6 = 2/5b (10b)
3 = 2b
log2 3 = b
a = .156 b = 1.585
y = .156x1.585
Practice Problems:
Write an exponential function of the form y=ab^x that passes through the given points.
1. (1,4), (2,12)
Write a power function of the form y=ax^b that passes through the given points.
1. (5,12), (7,25)
8.8 Logistic Growth Functions (pg. 517 - 522)
P(3) = 5.808089324
Practice Problems:
Graph and identify asymptotes, y-intercept and point of maximum.
1. y = 5/1+e^-10x
2. y = 8/1+e^-1.02x
Solve.
1. 10/1+2e^-4x= 9
2. 9/1+5e^-0.2x= 3/4
WEBSITES USED:
http://people.richland.edu/james/lecture/m116/logs/properties.html
http://www.mathwords.com/c/change_of_base_formula.htm
http://www.purplemath.com/modules/solvexpo.htm
Practice Answers:
8.1
1. 1; x-axis
2. 3/2; x-axis
1. domain; all real numbers. range; y>0
2. domain;all real numbers. range; y>1
8.2
1. decay
2. growth
1. domain; all real numbers. range; y>0
2. domain; all real numbers. range; y>-2
8.3
1. 3e^5
2. 64/e^6
1. 5.474
2. -0.199
8.4
1. 3
2. -0.38
1. x
1. y= e^x/6
8.5
1. 1.277
2. 1.226
8.6
1. 1
2. approx. .3495
3. 1
8.7
1. y=(4/3)3^x
1. y=.358x^2.181
8.8
1. Asymptotes- x-axis, y=5 ; y-int. (0,5/2) ; pt of max- (0, 5/2)
2. Asymptotes- x-axis, y=8 ; y-int. (0,4) ; pt of max- (0,4)
1. ln18/4, about .723
2. -3.942